Theoretical results from optimal control theory and numerical analysis play an important role in understanding and numerically solving optimal control problems with higher order state inequality constraints. However, the application of these theoretical results to numerical practice can sometimes be misleading. In this paper we examine a specific example which arose in the development of an industrial grade optimal control package that uses a direct transcription approach. This example illustrates several aspects of direct transcription codes that require a reinterpretation of the numerical and analytical theory.

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Direct transcription methods for the numerical solution of optimal control problems have the advantage that they do not require estimates of the adjoint variables. However, it is natural to want to use the discrete NLP multipliers to estimate the adjoint variables. It has been shown earlier in the literature for a large collection of numerical discretizations that order of the state and control variables found are generally independent of the implementation of the chosen discretization if no post processing is used to find the control. This is not always true for the adjoint estimation problem. The compressed trapezoidal discretization is used in some commercial codes. In this paper we show that the second order adjoint estimate result for the uncompressed trapezoidal method does not hold for the compressed trapezoidal method. We also show how to recover the lost order and carefully analyze convergence. Some related results are also discussed.

In the direct transcription approach to the numerical solution of optimal control problems, the optimal control problem is discretized and the resulting nonlinear programming problem is solved numerically. There has been considerable study over the last 10 years on order of convergence of cost, state, multipliers, and control. This paper discusses these questions, and the highly technical results in the literature, in the context of their implications for industrial grade optimal control packages.